### Abstract

A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Göttsche and Hirschowitz prove that on P^{2} every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef.

Original language | English (US) |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 81-104 |

Number of pages | 24 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 712 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Contemporary Mathematics*(pp. 81-104). (Contemporary Mathematics; Vol. 712). American Mathematical Society. https://doi.org/10.1090/conm/712/14343

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*Contemporary Mathematics.*Contemporary Mathematics, vol. 712, American Mathematical Society, pp. 81-104. https://doi.org/10.1090/conm/712/14343

**Weak brill-noether for rational surfaces.** / Coskun, Izzet; Huizenga, Jack.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Weak brill-noether for rational surfaces

AU - Coskun, Izzet

AU - Huizenga, Jack

PY - 2018/1/1

Y1 - 2018/1/1

N2 - A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Göttsche and Hirschowitz prove that on P2 every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef.

AB - A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Göttsche and Hirschowitz prove that on P2 every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef.

UR - http://www.scopus.com/inward/record.url?scp=85052248309&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052248309&partnerID=8YFLogxK

U2 - 10.1090/conm/712/14343

DO - 10.1090/conm/712/14343

M3 - Chapter

AN - SCOPUS:85052248309

T3 - Contemporary Mathematics

SP - 81

EP - 104

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -